A symmetry result for strictly convex domains

Abstract

Assume that $D\subset {\mathbb{R}}^2$ is a strictly convex domain with $C^2-$smooth boundary. {\bf Theorem.} {\em If ${\int }_D{e}^{ix}{y}^{n}dxdy=0$ for all sufficiently large $n$, then $D$ is a disc.}